J. Nig. Soc. Phys. Sci. 4 (2022) 251–264

Journal of the
Nigerian Society

of Physical
Sciences

Jackknife Kibria-Lukman M-Estimator: Simulation and
Application

Segun L. Jegedea,∗, Adewale F. Lukmanb,c, Kayode Ayinded, Kehinde A. Odeniyie

aDepartment of Mathematical Sciences, Kent State University
bDepartment of Mathematics, University of Medical Sciences, Nigeria

cInstitute of Mathematics, Eotvos Lorand University, Hungary
dDepartment of Statistics, Federal University of Technology, Akure, Nigeria

eDepartment of Agricultural Economics and Agribusiness Management, Osun State University, Nigeria

Abstract

The ordinary least square (OLS) method is very efficient in estimating the regression parameters in a linear regression model under classical
assumptions. If the model contains outliers, the performance of the OLS estimator becomes imprecise. Multicollinearity is another issue that
can reduce the performance of the OLS estimator. This study proposed the Robust Jackknife Kibria-Lukman (RJKL) estimator based on the
M-estimator to deal with multicollinearity and outliers. We examine the superiority of the estimator over existing estimators using theoretical
proofs and Monte Carlo simulations. We put the estimator to the test once more using real-world data. We observed that the estimator performs
better than the existing estimators.

DOI:10.46481/jnsps.2022.664

Keywords: Jackknife Kibria-Lukman, M-estimator, Monte Carlo Simulation, Multicollinearity, Outliers, Robust

Article History :
Received: 17 February 2022
Received in revised form: 21 March 2022
Accepted for publication: 24 March 2022
Published: 29 May 2022

c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: J. Ndam

1. Introduction

The regression model is commonly used in many disciplines
to analyze data. The model’s analysis is a form of predictive
modelling technique which statistically examines the relation-
ship between two different sets of variables. The first vari-
able is called the dependent, also known as the target variable.
The second variable is called the independent, also known as
the predictors because they are usually more than one. The
model is frequently used for forecasting. Mathematically, the

∗Corresponding author tel. no:
Email address: sjegede@kent.edu (Segun L. Jegede )

general regression model includes; an n × 1 vector of obser-
vations referred to as dependent variable labelled y, a known
full column rank of n × p standardize and centered independent
variables labelled X, a p × 1 vector of unknown parameters la-
belled β and an n × 1 vector of disturbances labelled ε. ε is as-
sumed to be normally distributed with E (ε) = 0 and dispersion
matrix Cov (ε) = σ2 I. The model is mathematically written as

y = Xβ + ε (1)

The Ordinary Least Square (OLS) method is very efficient in es-
timating the regression parameters in a linear regression model
under classical assumptions. The Gauss Markov theorem es-
tablishes this fact. The theorem stated the OLS estimator has
the best linear unbiased estimator (BLUE) with minimum vari-

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Jegede et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 251–264 252

ance in the class of all unbiased linear estimators. However, if
the dataset for the regression analysis contains outliers, the per-
formance of the OLS estimator becomes imprecise [1–3]. The
OLS estimator of β is given by

β̂ = S −1 X
′

y (2)

where S = X
′

X.
When outliers are present, robust regression is used, which gives
better results than the OLS method[4–7]. The M-estimation ap-
proach is the most common robust regression method which is
used to handle outlier in the y-direction [8]. It is a generaliza-
tion to maximum likelihood estimation in context of location
models. The means that it is nearly as efficient as the OLS.
The approach involves minimizing residual function rather than
minimizing the number of squared errors as the objective. Gen-
erally, the approach considers the likelihood function of β and

L (β,σ) =
n∏

i=1

1
σ

f
(

yi − xi
′

β

σ

)
=

1
σn

n∏
i=1

f
(

yi − xi
′

β

σ

)
, (3)

and by replacing the OLS criterion with a robust criterion, M-
estimator of β is

β̂M = minβ
n∑

i=1

ρ

(
yi − xi

′

β

σ̂M

)
(4)

The purpose of this study is to propose an estimator that solves
the problem of multicollinearity and outlier in linear regression
model. We investigate the superiority of the estimator through
theoretical comparison, simulations and practical application.

2. Existing Shrinkage Estimators

A popular shrinkage estimator is the ridge estimator (RE), de-
veloped by [9] and expressed as:

β̂ K = (S + kI)
−1 X

′

y = W(k)β̂ (5)

where W(k) = (S + kI)−1S and k > 0. However, RE can be
sensitive to outliers in the y-direction. Silvapulle [10] com-
bined the advantage of the ridge estimator and the M-estimator
to form the Ridge M-estimator (RME), expressed as follows:

β̂ M K = W(k)β̂ M, (6)

Kibria and Lukman [11] recently proposed an estimator called
the Kibria-Lukman (KL) estimator, the estimator is expressed
as:

β̂ K L = (S + kI)
−1 (S − kI) β̂ = M(k)β̂ (7)

where M (k) = (S + kI)−1 (S − kI) =
(
I − 2k(S + kI)−1

)
.

Grafting the M-estimator into the KL estimator produced the
robust KL estimator as follows:

β̂ M K L = M(k)β̂ M, (8)

The use of Jackknife is also a popular approach for reducing the
biasness in bias estimator proposed for handling multicollinear-
ity [12–14]. Ugwuowo et al. [15] proposed a jackknife KL

(JKL) estimator under the proposition that the use of jackknife
procedure reduces the bias of the ridge estimator [16]. The JKL
estimator is expressed as:

β̂ JK L =
(
I − 2k(S + kI)−1

)2 (
I + 2k(S + kI)−1

)
β̂

= (M(k))2 N(k)β̂ (9)

where N (k) = (S + kI)−1 (S + 3kI) =
(
I + 2k(S + kI)−1

)
.

2.1. A New Robust Estimator
Using the same approach as [10,17,3], we combined the JKL
estimator in (9) and the M-estimator to form the Robust Jack-
knife Kibria-Lukman (RJKL) estimator. It is obvious that the
presence of outliers in the y-direction will reduced the effi-
ciency of the JKL estimator. Thus, we defined the RJKL es-
timator as follows

β̂ RJKL = (M(k))
2 N(k)β̂ M (10)

where k > 0.
The canonical form of model (1) is written as

Y = Zα + ε, (11)

where Z = XT, α = T′β and T is the ortogonal matrix whose
columns contains the eigenvectors of X′X. Then

Z
′

Z = T
′

X
′

XT = Λ = diag
(
λ1, λ2, . . . , λp

)
, (12)

where λ1, λ2, . . . , λp > 0 are the ordered eigenvalues of X′X.
Let α̂M be an M-defined by the solution of the M-estimating
equations

∑
ϕ(ei/s)zi = 0 where ei = yi − ziα̂M, s is an esti-

mator of scale for the errors and ϕ(.) is some suitably chosen
function [18]. Thus, the estimators presented in (2-9) can be
written in canonical form as follows:

α̂ = Λ−1Z
′

y (13)

α̂K = (Λ+kI)
−1Z

′

y = W∗(k)α̂ (14)

α̂M K = W
∗(k)α̂M (15)

α̂K L =
(
I − 2k(Λ + kI)−1

)
α̂ = M∗(k)α̂ (16)

α̂M K L = M
∗(k)α̂M (17)

α̂JK L =
(
I − 2k(Λ + kI)−1

)2 (
I + 2k(Λ + kI)−1

)
α̂

= (M∗(k))2 N∗(k)α̂ (18)

α̂RJK L = (M
∗ (k))2 N∗ (k) α̂M (19)

where W∗(k) = Λ (Λ+kI)−1, M∗(k) =
(
I − 2k(Λ + kI)−1

)
,

N ∗(k) =
(
I + 2k(Λ + kI)−1

)
for k>0.

The organization of this article is as follows. The theoretical
comparison among estimators is given in section 2.2. Robust
choice of the biasing parameters are discussed in section 3, a
simulation study conducted to evaluate the performance of the
proposed estimator in section 4 and a real life data was analyzed
in Section 6 to illustrate the finding of the paper. Section 7 ends
with some concluding remarks.

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Jegede et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 251–264 253

2.2. Superiority of the RJKL Estimator
The Bias of an estimator β̃ is expressed in equation (20), its
Mean Squared Error Matrix (MSEM) in equation (21) and its
Scalar Mean Squared Error (MSE) in equation (22).

Bias
(
β̃
)

= E
(
β̃
)
−β (20)

MS E M
(
β̃
)

= E
(
β̃−β

) (
β̃−β

)′
= D

(
β̃
)

+ Bias
(
β̃
)

Bias
(
β̃
)′

(21)

MS E
(
β̃
)

= E
(
β̃−β

) (
β̃−β

)′
= tr

(
D

(
β̃
)

+ Bias
(
β̃
)

Bias
(
β̃
)′)

(22)

where D
(
β̃
)

is the variance matrix of β̃, E
(
β̃
)

is the expecta-

tion of β̃ and tr (A) is the trace of a matrix, A. Also, for α̃ =
Tβ̃, MS E M (α̃) = T′MS E M

(
β̃
)

T and MS E (α̃) = MS E
(
β̃
)
.

Additionally, for two estimators β̃1 and β̃2, β̃1 is said to be su-
perior to β̃2 with respect to the MSEM criterion, if and only if,
MS E M

(
β̃1

)
−MS E M

(
β̃2

)
≥ 0. If MS E M

(
β̃1

)
−MS E M

(
β̃2

)
≥

0 then MS E
(
β̃1

)
− MS E

(
β̃2

)
≥ 0 for j = 1, 2. The converse

is not true. We also made use of the following lemmas for the
theoretical comparison:

Lemma 2.1 [19] For some vector, α and a positive definite ma-
trix, A (that is, A > 0); A −αα

′

≥ 0 if and only if αA−1α
′

≤ 1.

Lemma 2.2 [20] Let β̃1 and β̃2 be two competing estimators of
β. Suppose that the difference between the covariance of the
two estimators, D = Cov

(
β̃1

)
− Cov

(
β̃2

)
> 0, then ∆

(̂
β1, β̂2

)
=

MS E M
(
β̃1

)
−MS E M

(
β̃2

)
≥ 0 if and only if d

′

2(D + d1d
′

1)
−1

d1 ≤

1. MS E M
(
β̃ j

)
and d j denote the mean squared error matrix and

bias vector of β̃ j respectively for j = 1, 2.

We used the MSE to prove the superiority of the RJKL. The
Mean Squared Error of the OLS estimator and the M-Estimator
is expressed in equation (23) and (24), respectively.

MS E
(
α̂
)

= σ2
p∑

i=1

1
λi

(23)

MS E
(
α̂M

)
=

p∑
i=1

Ωii (24)

where Ωii = Cov
(
α̂M

)
.

The mean squared error matrix and the scalar mean squared
error of the JKL is defined as follows:

MS E M
(
α̂JK L

)
= σ̂2

(
I − (2k(Λ + kI)−1)

2
)2

Λ
−1

(
I − 2k(Λ + kI)−1

)2
+ Bias

(
α̂JK L

)′
Bias

(
α̂JK L

)
(25)

MS E
(
α̂JK L

)
= σ̂2

p∑
i=1

(
(λi + k)

2
− 4k2

)2
(λi − k)

2

λi(λi + k)
6

+

p∑
i=1

(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2
α2i

(λi + k)
6

(26)

where
Bias

(
α̂JK L

)
=

[(
I − 2k(Λ + kI)−1

)2 (
I + 2k(Λ + kI)−1

)
− I

]
α.

The corresponding mean square error matrix and the scalar mean
square error for the RJKL estimator is:

MS E M
(
α̂RJK L

)
=

(
I − (2k(Λ + kI)−1)

2
)2

Ω
(
I + 2k(Λ + kI)−1

)2
+ Bias

(
α̂JK L

)′
Bias

(
α̂JK L

)
(27)

MS E
(
α̂RJK L

)
=

p∑
i=1

Ωii
(
(λi + k)

2
− 4k2

)2
(λi − k)

2

(λi + k)
6

+

p∑
i=1

(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2
α2i

(λi + k)
6

(28)

We only consider the JKL and the robust estimators in the the-
oretical comparison and we impose the following conditions to
present the main theorems:

1. ϕ is skew-symmetric and non-decreasing;
2. The errors are symmetric;
3. O is finite.

2.2.1. Superiority of the RJKL estimator over the JKL estima-
tor

We state a theorem for RJKL to be superior to the JKL estima-
tor:

Theorem 2.1: The proposed estimator, α̂RJK L is superior to the
jackknife Kibria Lukman estimator, α̂JKL if and only if then
O < σ2Λ−1 for every k > 0.

Proof: The difference between the MSE of the RJKL and the
JKL estimator from (28) and (26) is

∆1 = MS E
(
α̂RJK L

)
− MS E

(
α̂JKL

)
=

p∑
i=1

Ωii
(
(λi + k)

2
− 4k2

)2
(λi − k)

2

(λi + k)
6

−
σ̂2

(
(λi + k)

2
− 4k2

)2
(λi − k)

2

λi(λi + k)
6

=

p∑
i=1

(
Ωii −

σ2

λi

) 
(
(λi + k)

2
− 4k2

)2
(λi − k)

2

(λi + k)
6

 (29)
For ∆1 < 0 will imply that

∑p
i=1 Ωii <

∑p
i=1 σ

2/λi, k > 0.
253



Jegede et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 251–264 254

2.2.2. Superiority of the RJKL estimator over the robust KL estimator
The scalar mean square error of the KL estimator is

MS E(α̂M KL) =
p∑

i=1

(λi − k)
2

(λi + k)
2
Ωii +

p∑
i=1

4α2i k
2

(λi + k)
2

(30)

Theorem 2.2: The proposed estimator, α̂RJK L is superior to the robust Kibria Lukman estimator, α̂M KL if and only if

αΩ−1
[
2
(
I − 2k(Λ + kI)−1

)−1((
I − 2k(Λ + kI)−1

) (
I + 2k(Λ + kI)−1

)
+ 1

)−1
− I

]
α
′

< 1

for k > 0.

Proof: The difference between the MSE of the RJKL and the KL estimator from (28) and (30) is

∆2 = MS E
(
α̂RJKL

)
− MS E

(
α̂MK L

)
=

p∑
i=1

Ωii(λi − k)
2

(λi + k)
2


(
(λi + k)

2
− 4k2

)2
(λi + k)

4
− 1


+

p∑
i=1

α2i

(λi + k)
2


(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2

(λi + k)
4

− 4k2

 (31)
Consequently,

Ω

[
2
(
I − 2k(Λ + kI)−1

)−1((
I − 2k(Λ + kI)−1

) (
I + 2k(Λ + kI)−1

)
+ 1

)−1
− I

]−1
is positive definite, provided that

(
(λi + k)

2
− 4k2

)2
> (λi + k)

4 and
(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2
> 4k2 (λi + k)

4
.

2.2.3. Superiority of the RJKL estimator over the Ridge M-estimator
The scalar mean square error of the Ridge M-estimator is

MS E(α̂M (k)) =
p∑

i=1

λi
2

(λi + k)
2
Ωii +

p∑
i=1

k2α2i
(λi + k)

2
(32)

Theorem 2.3: The proposed estimator, α̂RJK L is superior to the Ridge M-estimator, α̂M K if and only if

αΩ−1
[((

I − 2k(Λ + kI)−1
)2 (

I + 2k(Λ + kI)−1
)
− I

)2
−

((
I − k(Λ + kI)−1

)
− I

)2]
[(

I − k(Λ + kI)−1
)2
−

(
I − (2k(Λ + kI)−1)

2
)2(

I + 2k(Λ + kI)−1
)2]−1

α
′

< 1

for k > 0.
Proof: The difference between the MSE of the RJKL and the Ridge M-estimator from (28) and (32) is

∆3 = MS E
(
α̂RJKL

)
− MS E

(
α̂M K

)
=

p∑
i=1

Ωii

(λi + k)
2


(
(λi + k)

2
− 4k2

)2
(λi − k)

2

(λi + k)
4

−λi
2


+

p∑
i=1

α2i

(λi + k)
2


(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2

(λi + k)
4

− k2

 (33)
Consequently,

Ω

[((
I − 2k(Λ + kI)−1

)2 (
I + 2k(Λ + kI)−1

)
− I

)2
−

((
I − k(Λ + kI)−1

)
− I

)2]−1
[(

I − k(Λ + kI)−1
)2
−

(
I − (2k(Λ + kI)−1)

2
)2(

I + 2k(Λ + kI)−1
)2]

is positive definite, provided that
(
(λi + k)

2
− 4k2

)2
(λi − k)

2 > λi
2 (λi + k)

4
and

(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2
> k2 (λi + k)

4
.

254



Jegede et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 251–264 255

2.2.4. Superiority of the RJKL estimator over the M-estimator
We state a theorem for RJKL to be superior to the M-estimator:

Theorem 2.4: The proposed estimator, α̂RJK L is superior to the
M-estimator, α̂M if and only if

αΩ−1
(
I −

(
I −

(
2k(Λ + kI)−1

)2)2(
I + 2k(Λ + kI)−1

)2)−1
((

I − 2k(Λ + kI)−1
)2 (

I + 2k(Λ + kI)−1
)
− I

)2
α
′

< 1

for k > 0.

Proof: The difference between the MSE of the RJKL and the
M estimator from (28) and (24) is

∆4 = MS E
(
α̂RJK L

)
− MS E

(
α̂M

)
=

p∑
i=1

Ωii


(
(λi + k)

2
− 4k2

)2
(λi − k)

2

(λi + k)
6

− 1


+

p∑
i=1

(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2
α2i

(λi + k)
6

(34)

Consequently,

Ω

(
I −

(
I −

(
2k(Λ + kI)−1

)2)2(
I + 2k(Λ + kI)−1

)2)
((

I − 2k(Λ + kI)−1
)2 (

I + 2k(Λ + kI)−1
)
− I

)2−1
is positive definite, provided that

(
(λi + k)

2
− 4k2

)2
(λi − k)

2 >

(λi + k)
6. Notice that the right hand expression after the addi-

tion sign is positive, this follows from bias squared.

3. Robust choice of the biasing parameter

It is customary to use the optimization procedure to obtain the
biasing parameter of an estimator [3]. This is done by mini-
mizing equation (28), which is rewritten in (35), with respect to
k.

f (k) =
p∑

i=1

Ωii
(
(λi + k)

2
− 4k2

)2
(λi − k)

2

(λi + k)
6

+

p∑
i=1

(
(λi − k)

2 (λi + 3k) − (λi + k)
3
)2
α2i

(λi + k)
6

(35)

This can be obtained by setting

∂ f (k)
∂k

= 0 (36)

Proceeding with (36) will yield a rather complex estimation for
k. Thus, we propose to use the robust version of the biasing

parameter used for the jackknife KL estimator [15]. The biasing
parameter used for the jackknife KL estimator is presented in
(37).

k̂ =

√
pσ̂2∑p
i=1 α̂

2
i

(37)

This parameter (37) is the squared root of harmonic mean of the
biasing parameter used for ridge estimator. The robust equiva-
lence of the parameter, k̂ is

k̂M =

√
pÂ2∑p

i=1 α̂
2
Mi

(38)

where Â2 is given by Huber [8] as

Â2 =
s2(n − p)−1

∑p
i=1 (ϕ(ei/s))

2

(
∑p

i=1 (1/n)ϕ
′(ei/s))

2
(39)

With the assumption that α̂M ∼ N(α,A2Λ−1). And, this holds
since n

1
2

(
α̂2M −α

)
ΛN

(
0, A2Λ−1

)
, where

A2 =
s2o E(ϕ

2(ε/so))

(E(ϕ′(ε/so)))
2
, (40)

with the scale estimate so.

4. Monte Carlo Simulation Study

We adopt the Monte Carlo simulation design [21, 22] to observe
the superiority of the Robust Jackknife KL estimator. The de-
sign was also recently adopted in related studies [23–28]. R
programming language was used for the simulation.
The following equation is used to generate the predictors:

xi j = (1 −ρ
2)

1/2
zi j + ρzi,p+1,

i = 1, 2, . . . , n, j = 1, 2, . . . , p (41)

where ρ2 denotes the correlation between independent variables
and zi j are pseudo-random numbers from the standard normal
distribution. The coefficients β1,β2, . . . , βp are selected as the
normalized Eigenvectors corresponding to the largest eigenvalue
of X

′

X such that β
′

β = 1. This is a common restriction in sim-
ulation studies of this kind [3, 25–26, 29–30].
The dependent variable determined by

yi = β0 + β1 xi1 + β2 xi2 + · · · + βp xip + εi,

i = 1, 2, . . . , n (42)

where the ε′i s are independently generated from N(0, σ
2). We

consider number of independent variables, p = 3 and p = 7. We
introduced 10%, 20% and 30% outlier into each sample size
considered in the simulation study [31]. The other specifica-
tions considered in the simulation design is as follows:

1. ρ = 0.70, 0.80, 0.90, 0.99
2. σ = 1, 5, 10

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Jegede et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 251–264 256

Table 1: Estimated MSEs of the estimators when there are no outliers and p=3

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 0.0906 0.0945 0.0817 0.0856 0.0743 0.0781 0.0640 0.0675

0.8 0.1201 0.1250 0.1033 0.1082 0.0886 0.0935 0.0663 0.0708
0.9 0.2145 0.2232 0.1633 0.1720 0.1201 0.1286 0.0618 0.0689
0.99 1.9546 2.0313 0.5968 0.6575 0.0671 0.0920 0.0373 0.0400

5 0.7 2.2644 2.3616 1.7243 1.8253 1.2825 1.3828 0.7235 0.8078
0.8 3.0013 3.1259 2.1305 2.2606 1.4436 1.5713 0.6654 0.7631
0.9 5.3623 5.5804 3.2613 3.4870 1.7660 1.9747 0.5260 0.6413
0.99 48.8650 50.7823 13.0782 14.5443 2.0620 2.4665 5.4359 5.1779

10 0.7 9.0574 9.4463 6.6716 7.0819 4.7377 5.1484 2.3424 2.6843
0.8 12.0052 12.5036 8.2686 8.7944 5.3753 5.8912 2.2291 2.6120
0.9 21.4491 22.3218 12.7356 13.6432 6.6773 7.5085 1.8716 2.3134
0.99 195.4598 203.1294 52.0364 57.8922 8.4338 10.0135 23.8768 22.6897

100 1 0.7 0.0470 0.0492 0.0443 0.0464 0.0418 0.0438 0.0376 0.0395
0.8 0.0637 0.0666 0.0583 0.0611 0.0533 0.0560 0.0445 0.0470
0.9 0.1159 0.1213 0.0985 0.1034 0.0826 0.0871 0.0562 0.0600
0.99 1.0714 1.1209 0.4493 0.4816 0.1042 0.1212 0.0088 0.0096

5 0.7 1.1756 1.2303 0.9732 1.0242 0.7955 0.8426 0.5246 0.5635
0.8 1.5919 1.6658 1.2475 1.3150 0.9527 1.0137 0.5344 0.5815
0.9 2.8982 3.0318 2.0106 2.1281 1.3059 1.4063 0.5010 0.5642
0.99 26.7861 28.0236 9.3727 10.1409 1.7215 2.0322 0.7853 0.7860

10 0.7 4.7023 4.9212 3.7496 3.9546 2.9304 3.1197 1.7457 1.8980
0.8 6.3674 6.6630 4.8106 5.0824 3.5146 3.7596 1.7988 1.9809
0.9 11.5929 12.1274 7.7879 8.2604 4.8593 5.2603 1.7529 1.9918
0.99 107.1445 112.0945 37.1706 40.2385 6.9026 8.1198 4.0429 3.9826

200 1 0.7 0.0229 0.0240 0.0222 0.0233 0.0216 0.0226 0.0205 0.0215
0.8 0.0310 0.0325 0.0297 0.0311 0.0284 0.0298 0.0260 0.0273
0.9 0.0566 0.0593 0.0520 0.0545 0.0475 0.0499 0.0394 0.0415
0.99 0.5249 0.5498 0.2979 0.3151 0.1365 0.1474 0.0149 0.0181

5 0.7 0.5723 0.6000 0.5105 0.5363 0.4536 0.4775 0.3558 0.3763
0.8 0.7761 0.8133 0.6660 0.6999 0.5659 0.5968 0.3999 0.4251
0.9 1.4158 1.4833 1.1108 1.1697 0.8468 0.8978 0.4615 0.4986
0.99 13.1226 13.7447 5.9327 6.3356 1.8403 2.0648 0.1855 0.2230

10 0.7 2.2894 2.4000 1.9666 2.0687 1.6748 1.7687 1.1970 1.2755
0.8 3.1043 3.2531 2.5609 2.6958 2.0798 2.2015 1.3328 1.4300
0.9 5.6634 5.9333 4.2718 4.5066 3.1064 3.3082 1.5428 1.6854
0.99 52.4905 54.9787 23.3878 24.9955 7.1815 8.0669 0.9502 1.0706

250 1 0.7 0.0185 0.0196 0.0180 0.0191 0.0176 0.0186 0.0168 0.0178
0.8 0.0250 0.0265 0.0241 0.0256 0.0232 0.0246 0.0216 0.0229
0.9 0.0456 0.0484 0.0424 0.0450 0.0393 0.0418 0.0337 0.0358
0.99 0.4227 0.4494 0.2553 0.2738 0.1308 0.1426 0.0200 0.0236

5 0.7 0.4614 0.4892 0.4180 0.4438 0.3775 0.4014 0.3062 0.3266
0.8 0.6248 0.6630 0.5462 0.5810 0.4738 0.5055 0.3503 0.3760
0.9 1.1391 1.2097 0.9160 0.9777 0.7197 0.7730 0.4200 0.4584
0.99 10.5672 11.2344 5.0044 5.4380 1.6925 1.9306 0.1565 0.1963

10 0.7 1.8457 1.9567 1.6107 1.7131 1.3959 1.4901 1.0355 1.1138
0.8 2.4993 2.6521 2.0988 2.2374 1.7399 1.8648 1.1655 1.2645
0.9 4.5565 4.8390 3.5136 3.7599 2.6261 2.8375 1.3832 1.5302
0.99 42.2689 44.9375 19.6719 21.4040 6.5464 7.4852 0.7302 0.8829

3. n = 50, 100, 200, 250

The biasing parameter, k̂ was used for the ridge, KL and Jack-
knife KL estimator while the robust version of the biasing pa-

rameter, k̂M is used for the robust version of the estimators, in-
cluding the robust jackknife KL. The simulation was done 2000
times by generating new pseudo-random numbers and the esti-

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Table 2: Estimated MSEs of the estimators when there is 10% outlier and p=3

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 13.6963 0.1479 10.0387 0.1313 7.0596 0.1169 3.3265 0.0949

0.8 19.9972 0.1966 13.7140 0.1671 8.8318 0.1409 3.4809 0.1002
0.9 39.1477 0.3526 23.1621 0.2666 12.0224 0.1943 3.1734 0.0968
0.99 384.1922 3.2231 101.8056 1.0736 15.5236 0.1687 43.3645 0.0674

5 0.7 137.1126 3.6961 99.5904 2.8692 69.2274 2.1761 31.7675 1.2357
0.8 184.4161 4.9145 125.4914 3.5931 80.0792 2.5253 31.1814 1.2177
0.9 334.6314 8.8142 196.9320 5.6335 101.7029 3.2920 27.0002 1.0943
0.99 3091.902 80.5657 819.9766 24.6852 126.6444 4.3096 359.7422 6.3585

10 0.7 515.965 14.7841 374.3237 11.2446 259.8384 8.3034 119.0229 4.3970
0.8 689.075 19.6578 468.3979 14.1016 298.5428 9.6738 116.2113 4.4187
0.9 1239.973 35.2564 729.1771 22.1921 376.3083 12.7275 100.0509 4.1134
0.99 11372.96 322.2612 3017.1655 98.4042 465.9051 17.3591 1325.6418 27.3046

100 1 0.7 7.6998 0.0710 6.0621 0.0665 4.6620 0.0622 2.6673 0.0549
0.8 11.3464 0.0960 8.4537 0.0872 6.0668 0.0790 2.9758 0.0645
0.9 22.3993 0.1742 14.8158 0.1462 9.0488 0.1209 3.1156 0.0796
0.99 221.3106 1.6064 75.0130 0.6830 13.3166 0.1701 9.6412 0.0139

5 0.7 79.8287 1.7757 62.0647 1.4746 46.9970 1.2094 25.9263 0.8017
0.8 109.6894 2.3985 80.9090 1.8927 57.3702 1.4589 27.4833 0.8379
0.9 202.3623 4.3550 133.0403 3.0711 80.7291 2.0451 27.6453 0.8410
0.99 1888.093 40.1531 641.9309 14.9983 116.0079 3.2139 84.8115 1.0515

10 0.7 300.1889 7.1028 233.0442 5.7389 176.1599 4.5550 96.8446 2.8014
0.8 409.2441 9.5939 301.4447 7.3756 213.4082 5.5098 102.0062 2.9647
0.9 749.2885 17.4197 492.0093 12.0162 298.1691 7.7904 102.0787 3.0693
0.99 6946.875 160.6115 2359.3943 59.6581 427.1866 12.8245 315.4987 5.0330

200 1 0.7 4.4751 0.0355 3.7892 0.0344 3.1754 0.0333 2.1944 0.0314
0.8 6.5295 0.0480 5.3021 0.0457 4.2303 0.0434 2.6193 0.0394
0.9 12.9185 0.0872 9.5866 0.0793 6.8417 0.0719 3.2967 0.0584
0.99 129.3138 0.8062 56.8508 0.4555 17.1766 0.2086 2.6167 0.0263

5 0.7 46.0261 0.8880 38.4685 0.7913 31.7480 0.7021 21.1678 0.5494
0.8 62.9938 1.1988 50.6022 1.0295 39.8739 0.8757 24.0539 0.6218
0.9 116.2244 2.1794 85.6010 1.7212 60.5908 1.3247 28.8401 0.7453
0.99 1090.0300 20.1500 479.2827 9.6267 145.7462 3.4039 23.5782 0.4220

10 0.7 175.6885 3.5519 146.6835 3.0737 120.9005 2.6397 80.3387 1.9222
0.8 238.7105 4.7953 191.5810 3.9980 150.8015 3.2894 90.7374 2.1769
0.9 436.7494 8.7175 321.4645 6.6959 227.3712 4.9901 108.0523 2.6457
0.99 4062.8734 80.5996 1786.2549 38.1570 543.0836 13.4103 87.9686 1.8838

250 1 0.7 4.0422 0.0296 3.4584 0.0288 2.9318 0.0280 2.0741 0.0266
0.8 5.8499 0.0403 4.8094 0.0387 3.8919 0.0371 2.4794 0.0341
0.9 11.4636 0.0738 8.6490 0.0682 6.2987 0.0627 3.1569 0.0527
0.99 114.3099 0.6868 52.1028 0.4135 16.7411 0.2115 2.3191 0.0344

5 0.7 38.3098 0.7405 32.3421 0.6688 26.9940 0.6018 18.4195 0.4841
0.8 52.5104 1.0077 42.6718 0.8802 34.0728 0.7629 21.1009 0.5630
0.9 96.9599 1.8450 72.5000 1.4935 52.2625 1.1835 25.7424 0.7081
0.99 910.4138 17.1656 413.6027 8.6770 132.6110 3.3776 19.6279 0.4017

10 0.7 145.3148 2.9619 122.5263 2.5978 102.1118 2.2638 69.4115 1.6977
0.8 197.7114 4.0306 160.4884 3.4168 127.9780 2.8640 79.0100 1.9680
0.9 362.2294 7.3800 270.6038 5.8033 194.8606 4.4479 95.7777 2.4934
0.99 3377.0204 68.6613 1533.5258 34.3510 491.4155 13.2504 72.9195 1.7006

mated MSE calculated as:

mse
(
α̂
)

=
1

2000

2000∑
j=1

(
α̂i j −αi

)′ (
α̂i j −αi

)
(43)

As the standard deviation, σ and the degree of multicollinear-
ity, ρ increases, the mean square error, MSEs of the estima-
tors, α̂, α̂M, α̂K, α̂M K, α̂K L, α̂M K L, α̂JK L and α̂RJK L also
increases. The mean square error, MSEs of the estimators,

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Table 3: Estimated MSEs of the estimators when there are 20% outliers and p=3

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 27.4909 0.2521 20.2574 0.2197 14.3415 0.1908 6.8243 0.1449

0.8 40.3804 0.3364 27.9507 0.2809 18.2475 0.2316 7.4111 0.1545
0.9 79.5740 0.6035 47.8373 0.4495 25.5539 0.3206 7.1857 0.1496
0.99 790.303 5.5425 218.992 1.9246 35.8040 0.3346 86.3257 0.1074

5 0.7 272.653 6.3015 199.545 4.9528 140.154 3.8028 65.9791 2.1712
0.8 368.888 8.4073 253.726 6.2640 164.505 4.4998 66.6474 2.2272
0.9 672.473 15.0834 402.222 9.9382 213.812 6.0549 60.7675 2.1345
0.99 6245.01 138.519 1727.07 45.6565 287.645 8.3804 699.754 7.5586

10 0.7 1025.56 25.2056 749.801 19.5587 526.109 14.7806 247.809 8.1339
0.8 1377.33 33.6286 946.375 24.7504 613.041 17.5204 248.908 8.4380
0.9 2490.26 60.3325 1488.33 39.3541 790.911 23.7011 226.165 8.2671
0.99 22949.5 554.071 6349.96 182.212 1065.63 33.6397 2586.37 32.0533

100 1 0.7 15.1429 0.1200 11.8102 0.1110 8.9723 0.1026 4.9686 0.0879
0.8 22.3024 0.1624 16.4695 0.1456 11.6798 0.1300 5.5478 0.1027
0.9 43.9747 0.2953 28.8485 0.2439 17.4039 0.1979 5.7482 0.1243
0.99 433.363 2.7235 145.132 1.1828 24.3159 0.3201 20.1608 0.0265

5 0.7 156.829 2.9992 121.501 2.5008 91.5606 2.0585 49.7936 1.3663
0.8 215.022 4.0580 157.956 3.2314 111.340 2.5171 52.3434 1.4738
0.9 396.164 7.3797 259.142 5.3047 155.911 3.6269 51.5900 1.5822
0.99 3695.54 68.0707 1242.22 27.1764 212.261 6.7520 172.944 1.4096

10 0.7 593.815 11.9965 459.802 9.8262 346.290 7.9204 188.139 5.0142
0.8 808.576 16.2319 593.809 12.7085 418.459 9.7046 196.747 5.4576
0.9 1479.48 29.5183 967.986 20.9228 582.716 14.0706 193.291 5.9866
0.99 13717.4 272.281 4619.94 108.333 794.266 26.9217 641.662 6.2819

200 1 0.7 8.5590 0.0546 7.1615 0.0526 5.9191 0.0507 3.9641 0.0472
0.8 12.4987 0.0740 10.0274 0.0700 7.8863 0.0661 4.7286 0.0589
0.9 24.7452 0.1350 18.1275 0.1214 12.7229 0.1086 5.8933 0.0856
0.99 247.230 1.2522 106.201 0.6986 30.1372 0.3139 4.8016 0.0399

5 0.7 88.0770 1.3649 73.1962 1.2110 59.9960 1.0693 39.3377 0.8269
0.8 120.611 1.8505 96.2486 1.5846 75.2182 1.3437 44.4431 0.9475
0.9 222.578 3.3753 162.506 2.6714 113.635 2.0624 52.3102 1.1721
0.99 2086.82 31.2969 897.195 15.4657 255.893 5.8296 41.8475 0.7606

10 0.7 336.691 5.4597 279.756 4.7377 229.235 4.0805 150.121 2.9871
0.8 457.457 7.4019 365.064 6.2033 285.290 5.1345 168.490 3.4419
0.9 836.317 13.5011 610.777 10.4839 427.256 7.9212 196.828 4.3292
0.99 7773.24 125.185 3346.15 61.5152 957.602 23.0985 155.941 3.2643

250 1 0.7 7.6189 0.0448 6.4429 0.0434 5.3895 0.0421 3.7021 0.0396
0.8 11.067 0.0609 8.9921 0.0581 7.1774 0.0554 4.4397 0.0502
0.9 21.7733 0.1115 16.2260 0.1019 11.6384 0.0927 5.6469 0.0760
0.99 218.570 1.0377 97.8022 0.6138 30.2036 0.3058 4.1657 0.0496

5 0.7 72.4863 1.1186 60.8732 1.0056 50.4973 0.9004 33.9836 0.7165
0.8 99.6846 1.5225 80.5435 1.3247 63.8719 1.1432 38.9391 0.8365
0.9 184.767 2.7871 137.222 2.2539 98.0540 1.7852 47.2803 1.0714
0.99 1741.86 25.9362 780.620 13.4438 242.517 5.5131 34.7890 0.7911

10 0.7 274.650 4.4745 230.508 3.9307 191.072 3.4313 128.323 2.5835
0.8 374.584 6.0901 302.465 5.1794 239.672 4.3583 145.838 3.0239
0.9 687.970 11.1482 510.550 8.8300 364.476 6.8313 175.387 3.9233
0.99 6427.93 103.743 2876.86 53.4205 891.498 21.7854 128.508 3.2871

α̂, α̂M, α̂K, α̂M K, α̂K L, α̂M K L, α̂JK L and α̂RJK L decrease
as the sample size increases. As expected, the MSEs increases
as the outliers increases. This is what births the purpose of
our study, to propose a better estimator which handles outliers

jointly with multicollinearity by yielding a minimum MSE. As
we know it, the performance of the OLS is poor in the presence
of outliers and multicollinearity. This can be observed in Table
(1-8) and when there is no outlier (Table 1 & 5), the jackknife

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Table 4: Estimated MSEs of the estimators when there are 30% outliers and p=3

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 42.8128 1.1147 31.6134 0.9605 22.4115 0.8201 10.6056 0.5859

0.8 63.1076 1.5114 43.7884 1.2524 28.6259 1.0210 11.5133 0.6538
0.9 124.554 2.8334 75.1313 2.1200 40.1568 1.5214 10.9966 0.7209
0.99 1235.20 26.6215 344.127 11.0545 51.1102 2.7550 114.977 0.3498

5 0.7 425.609 22.2058 313.429 18.2857 221.616 14.8223 104.952 9.4545
0.8 577.241 30.1395 399.724 23.7643 260.936 18.2928 105.598 10.4202
0.9 1054.87 53.3573 635.995 37.9961 340.290 25.6981 94.3308 11.0380
0.99 9841.98 506.709 2750.33 201.981 405.678 48.7122 923.202 12.2529

10 0.7 1603.19 88.0314 1180.05 72.1510 834.108 58.1816 395.727 36.7551
0.8 2157.41 115.778 1493.33 90.7417 974.720 69.3911 395.734 39.1248
0.9 3908.96 210.583 2356.57 149.327 1261.65 100.545 351.986 43.0182
0.99 36170.2 1987.74 10123.2 789.441 1503.91 189.681 3415.795 50.4719

100 1 0.7 21.9449 0.2312 16.9722 0.2108 12.7585 0.1917 6.8909 0.1579
0.8 32.4753 0.3138 23.7871 0.2769 16.6932 0.2427 7.7505 0.1834
0.9 64.4363 0.5716 41.9240 0.4641 25.0097 0.3685 8.1158 0.2194
0.99 638.769 5.2860 211.730 2.3982 35.7890 0.7310 30.2949 0.0588

5 0.7 229.929 5.7788 176.924 4.8626 132.203 4.0421 70.5340 2.7288
0.8 315.839 7.8429 230.364 6.3371 160.890 5.0219 74.1414 3.0440
0.9 582.709 14.2855 378.129 10.5470 224.968 7.4702 72.9364 3.5126
0.99 5437.17 132.0971 1804.28 57.3248 306.655 16.8488 262.016 2.5864

10 0.7 870.874 23.1149 669.761 19.2563 500.176 15.8246 266.649 10.4209
0.8 1187.82 31.3710 865.995 25.1117 604.560 19.6900 278.519 11.6933
0.9 2175.94 57.1419 1411.80 41.8709 839.947 29.4090 272.570 13.6617
0.99 20172.5 528.373 6699.34 228.894 1142.26 67.2673 975.172 11.0599

200 1 0.7 12.3718 0.1028 10.2729 0.0983 8.4133 0.0940 5.5140 0.0860
0.8 18.2202 0.1403 14.5162 0.1312 11.3207 0.1225 6.6581 0.1065
0.9 36.2226 0.2572 26.3754 0.2276 18.3733 0.2001 8.3854 0.1516
0.99 362.358 2.3898 154.744 1.3440 43.7583 0.6179 7.0546 0.0898

5 0.7 127.504 2.5702 105.398 2.2789 85.8416 2.0101 55.4493 1.5485
0.8 175.938 3.5056 139.749 3.0109 108.592 2.5619 63.3166 1.8203
0.9 326.571 6.4269 237.550 5.1467 165.332 4.0331 75.3718 2.3770
0.99 3073.00 59.7253 1318.54 31.2958 377.572 13.1203 61.8303 1.9138

10 0.7 485.060 10.2807 400.873 8.9939 326.389 7.8142 210.598 5.8199
0.8 664.064 14.0223 527.384 11.8961 409.708 9.9837 238.676 6.8910
0.9 1221.52 25.7073 888.342 20.3805 618.079 15.7932 281.502 9.1218
0.99 11394.4 238.904 4885.92 124.855 1397.64 52.2367 229.319 7.8085

250 1 0.7 10.0629 0.0740 8.4145 0.0713 6.9507 0.0687 4.6546 0.0639
0.8 14.6414 0.1007 11.7454 0.0952 9.2400 0.0899 5.5572 0.0800
0.9 28.8334 0.1844 21.1552 0.1661 14.8922 0.1487 6.9958 0.1176
0.99 288.820 1.7173 125.332 1.0015 37.0075 0.4891 6.3407 0.0793

5 0.7 95.2388 1.8496 79.1862 1.6547 64.9614 1.4739 42.7579 1.1599
0.8 130.944 2.5165 104.553 2.1822 81.7941 1.8767 48.5731 1.3643
0.9 242.623 4.6092 177.371 3.7318 124.329 2.9620 57.9075 1.7933
0.99 2288.42 42.9196 991.234 22.8923 291.752 9.9008 51.2254 1.5034

10 0.7 361.592 7.3981 300.594 6.5134 246.530 5.6998 162.094 4.3143
0.8 493.526 10.0661 394.058 8.5973 308.269 7.2706 182.987 5.1045
0.9 907.090 18.4365 663.258 14.7336 465.033 11.5268 216.678 6.8007
0.99 8483.85 171.678 3677.92 91.2278 1084.86 39.3404 190.115 6.1180

KL estimator performs better than other non-robust estimator as
argued by Ugwuowo et al. [15]. The proposed estimator in this
study, α̂RJKL performs much better than the existing estimators
considered in this study. That is, the Robust Jackknife Kibria

Lukman estimator has the smallest mean square error.

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Table 5: Estimated MSEs of the estimators when there are no outliers and p=7

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 0.2697 0.2861 0.2208 0.2391 0.1776 0.1971 0.1106 0.1302

0.8 0.3781 0.4014 0.2898 0.3163 0.2144 0.2426 0.1084 0.1346
0.9 0.7106 0.7545 0.4639 0.5153 0.2738 0.3261 0.0779 0.1134
0.99 6.7379 7.1589 1.7016 2.0728 0.2011 0.2786 0.4034 0.2812

5 0.7 6.7424 7.1523 4.6574 5.1544 3.0057 3.5329 1.1284 1.5434
0.8 9.4532 10.0345 6.0780 6.7871 3.5309 4.2619 1.0342 1.5256
0.9 17.7638 18.8637 9.7972 11.1263 4.4276 5.6791 0.7544 1.3060
0.99 168.448 178.973 40.1434 49.2488 5.8421 7.1385 16.9292 11.8110

10 0.7 26.9695 28.6090 18.3182 20.3240 11.5532 13.6742 4.1293 5.7498
0.8 37.8129 40.1378 23.9652 26.8178 13.6418 16.5678 3.8478 5.7570
0.9 71.0552 75.4550 38.7965 44.1234 17.2632 22.2467 2.8872 5.0310
0.99 673.791 715.892 160.257 196.655 23.5174 28.6142 68.9781 48.1329

100 1 0.7 0.1398 0.1480 0.1250 0.1330 0.1112 0.1190 0.0868 0.0942
0.8 0.1966 0.2081 0.1687 0.1799 0.1431 0.1540 0.1000 0.1099
0.9 0.3704 0.3920 0.2855 0.3062 0.2123 0.2318 0.1072 0.1228
0.99 3.5231 3.7282 1.2434 1.3929 0.1913 0.2582 0.0273 0.0273

5 0.7 3.4947 3.6994 2.7012 2.9052 2.0214 2.2188 1.0551 1.2194
0.8 4.9144 5.2019 3.5908 3.8762 2.4962 2.7675 1.0860 1.2920
0.9 9.2599 9.8007 5.9888 6.5146 3.4971 3.9695 0.9865 1.2604
0.99 88.0776 93.2049 28.4307 32.0951 3.8710 5.2275 1.7745 1.5159

10 0.7 13.9790 14.7976 10.5672 11.3906 7.6909 8.4885 3.7718 4.4183
0.8 19.6575 20.8078 14.0815 15.2312 9.5407 10.6323 3.9301 4.7328
0.9 37.0396 39.2029 23.6031 25.7142 13.4985 15.3861 3.6637 4.7223
0.99 352.311 372.820 113.347 127.995 15.4177 20.7938 7.4482 6.3335

200 1 0.7 0.0647 0.0683 0.0614 0.0649 0.0582 0.0616 0.0522 0.0554
0.8 0.0909 0.0959 0.0844 0.0892 0.0782 0.0829 0.0667 0.0710
0.9 0.1710 0.1805 0.1498 0.1587 0.1300 0.1383 0.0955 0.1027
0.99 1.6255 1.7153 0.8128 0.8796 0.2906 0.3334 0.0165 0.0237

5 0.7 1.6178 1.7071 1.3832 1.4686 1.1688 1.2499 0.8092 0.8805
0.8 2.2718 2.3972 1.8662 1.9850 1.5036 1.6150 0.9280 1.0217
0.9 4.2758 4.5118 3.2115 3.4303 2.3108 2.5089 1.0747 1.2205
0.99 40.6370 42.8824 17.8112 19.4537 4.8583 5.8012 0.2790 0.3684

10 0.7 6.4711 6.8284 5.3885 5.7326 4.4155 4.7435 2.8485 3.1339
0.8 9.0871 9.5887 7.2798 7.7584 5.6921 6.1419 3.2789 3.6510
0.9 17.1032 18.0474 12.5815 13.4614 8.8149 9.6115 3.8625 4.4339
0.99 162.548 171.530 70.8341 77.4027 19.1209 22.8692 1.1435 1.4841

250 1 0.7 0.0507 0.0535 0.0486 0.0513 0.0466 0.0492 0.0427 0.0452
0.8 0.0712 0.0752 0.0671 0.0710 0.0632 0.0668 0.0557 0.0590
0.9 0.1341 0.1417 0.1205 0.1275 0.1076 0.1141 0.0843 0.0898
0.99 1.2751 1.3466 0.6942 0.7454 0.2948 0.3277 0.0261 0.0336

5 0.7 1.2667 1.3375 1.1070 1.1734 0.9588 1.0211 0.7022 0.7559
0.8 1.7806 1.8802 1.5009 1.5932 1.2466 1.3317 0.8261 0.8965
0.9 3.3536 3.5415 2.6044 2.7738 1.9560 2.1063 1.0100 1.1204
0.99 31.8780 33.6647 14.9915 16.2498 4.7865 5.5181 0.2975 0.3886

10 0.7 5.0670 5.3499 4.3067 4.5737 3.6140 3.8643 2.4612 2.6752
0.8 7.1226 7.5208 5.8442 6.2154 4.7030 5.0456 2.8969 3.1764
0.9 13.4145 14.1658 10.1797 10.8600 7.4268 8.0308 3.5881 4.0222
0.99 127.512 134.659 59.5520 64.5848 18.7782 21.6873 1.1794 1.5301

5. Real-Life Application

We adopted the Hussein data for the data analysis

yi = β0 +β1 xi1 + β2 xi2 +β3 xi3 +εi, i = 1, 2, . . . , 31(44)

Hussein and Zari [32] were the first to originally adopt the Hus-
sein data and was recently used to test the performance of a
proposed two parameter estimator in the presence of outlier [3].
The data contains 31 observations and three independent vari-

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Table 6: Estimated MSEs of the estimators when there is 10% outlier and p=7

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 60.8029 0.4791 40.8878 0.3938 25.4231 0.3180 8.8012 0.1997

0.8 99.2107 0.6718 62.2013 0.5205 34.8383 0.3907 9.5212 0.2056
0.9 216.044 1.2625 116.569 0.8524 50.8361 0.5308 8.2968 0.1780
0.99 2333.39 11.9803 545.305 3.6804 84.8065 0.5415 260.208 0.3678

5 0.7 422.505 11.9761 283.381 8.7943 175.556 6.1779 60.2840 2.8467
0.8 611.046 16.7925 382.392 11.6306 213.564 7.5504 57.9704 2.9104
0.9 1187.80 31.5545 640.163 19.2652 278.541 10.3922 45.1488 2.6870
0.99 11631.7 299.420 2716.67 89.2637 420.106 13.4739 1286.53 13.2986

10 0.7 1510.69 47.9040 1013.53 34.8725 628.169 24.2323 215.967 10.9305
0.8 2136.76 67.1685 1337.73 46.1804 747.614 29.6979 203.205 11.2630
0.9 4054.03 126.217 2185.96 76.6677 951.854 41.0706 154.244 10.5271
0.99 38783.8 1197.65 9062.67 356.691 1393.78 53.9159 4259.15 53.9040

100 1 0.7 33.3801 0.2180 24.8173 0.1937 17.6889 0.1711 8.2997 0.1315
0.8 54.6386 0.3065 38.4207 0.2612 25.3986 0.2199 9.9254 0.1510
0.9 119.500 0.5773 74.5251 0.4430 41.2638 0.3280 10.4418 0.1645
0.99 1282.08 5.4894 397.063 2.0845 50.4080 0.4107 34.2417 0.0428

5 0.7 239.390 5.4501 177.582 4.2958 126.232 3.2974 58.9316 1.8394
0.8 344.928 7.6607 242.290 5.7469 159.995 4.1420 62.5281 1.9869
0.9 668.299 14.4296 416.904 9.7203 231.095 6.0453 58.9708 2.0327
0.99 6539.09 137.201 2032.17 49.2646 262.700 8.9174 174.470 1.9351

10 0.7 873.665 21.8004 647.966 16.9514 460.494 12.7991 214.904 6.8875
0.8 1236.35 30.6427 868.354 22.7161 573.360 16.1320 224.127 7.5045
0.9 2347.62 57.7179 1464.59 38.5455 812.026 23.6971 207.553 7.8010
0.99 22517.5 548.797 7002.33 196.679 908.997 35.5307 602.934 7.9639

200 1 0.7 26.5675 0.1008 21.7406 0.0952 17.4507 0.0898 10.7326 0.0796
0.8 43.6640 0.1415 34.3492 0.1307 26.2661 0.1203 14.3672 0.1011
0.9 94.7433 0.2663 68.3657 0.2314 46.6883 0.1991 19.2553 0.1433
0.99 988.387 2.5302 417.755 1.3002 104.913 0.4966 6.8410 0.0385

5 0.7 138.754 2.5188 113.288 2.1673 90.6932 1.8448 55.4578 1.2999
0.8 200.219 3.5364 157.283 2.9357 120.070 2.3959 65.4601 1.5274
0.9 387.860 6.6543 279.733 5.1010 190.928 3.7714 78.7039 1.8875
0.99 3780.44 63.2284 1600.02 29.7487 403.397 9.6212 26.2144 0.6796

10 0.7 499.716 10.0752 407.821 8.5153 326.312 7.1013 199.297 4.7765
0.8 706.782 14.1453 555.039 11.5504 423.557 9.2456 230.727 5.6423
0.9 1341.19 26.6170 967.189 20.1369 660.061 14.6469 272.047 7.0741
0.99 12846.0 252.909 5439.26 118.575 1372.75 38.1181 89.1150 2.7126

250 1 0.7 23.7443 0.0798 19.9966 0.0761 16.6024 0.0726 11.0348 0.0658
0.8 39.3865 0.1121 32.0454 0.1051 25.5322 0.0983 15.3790 0.0855
0.9 86.3019 0.2113 65.0884 0.1882 47.1256 0.1665 22.3962 0.1279
0.99 926.147 2.0085 432.778 1.1122 136.614 0.4907 8.4626 0.0533

5 0.7 125.902 1.9932 105.646 1.7495 87.3520 1.5231 57.5402 1.1285
0.8 183.632 2.8026 149.001 2.3814 118.342 1.9971 70.8069 1.3553
0.9 359.783 5.2793 270.848 4.1691 195.660 3.1997 92.5738 1.7509
0.99 3548.57 50.1888 1656.48 25.1650 522.139 9.2833 32.6286 0.7798

10 0.7 445.692 7.9727 373.720 6.8642 308.752 5.8468 203.024 4.1248
0.8 633.414 11.2102 513.634 9.3527 407.647 7.6792 243.524 4.9686
0.9 1206.99 21.1167 908.186 16.4252 655.669 12.3754 309.832 6.5003
0.99 11598.9 200.749 5412.89 100.221 1705.35 36.7024 106.558 3.0723

ables. Details description can be found in [32,3].
The data contains multicollinearity with its variance infla-

tion factor, VIF>10 and about 19.4% outliers in the y-direction
at observations 12, 14, 15, 16, 30 and 31. Hence, it is suffi-

cient to use the data in this study. The output of the analysis is
presented in Table 9.
The RJKL estimator has the smallest mean square value. Thus,
the RJKL estimator performed better. The percentage MSE

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Table 7: Estimated MSEs of the estimators when there are 20% outliers and p=7

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 119.914 1.0249 80.5354 0.8315 50.0589 0.6608 17.4943 0.3986

0.8 196.356 1.4329 123.079 1.1009 69.0594 0.8176 19.2267 0.4192
0.9 428.611 2.9971 231.566 2.0723 101.677 1.3390 17.2449 0.5015
0.99 4636.04 29.4453 1092.23 10.7202 177.533 2.2999 545.556 0.5486

5 0.7 818.077 25.1608 547.263 19.0514 338.257 13.9238 116.863 7.0046
0.8 1184.59 35.2298 739.444 25.3346 412.225 17.3117 113.255 7.4746
0.9 2304.40 66.2106 1238.94 42.5563 539.034 24.8324 90.0158 7.5425
0.99 22578.0 629.137 5270.26 209.884 866.359 36.1990 2712.14 16.1937

10 0.7 2921.54 100.502 1955.11 75.7647 1209.05 55.0819 418.169 27.4365
0.8 4139.97 140.781 2585.75 100.868 1442.75 68.6166 397.063 29.4010
0.9 7866.74 264.728 4232.82 169.731 1843.98 98.7288 308.658 29.8633
0.99 75384.1 2517.78 17619.9 839.856 2884.57 145.069 8964.29 65.3513

100 1 0.7 64.3033 0.3636 47.2958 0.3187 33.2345 0.2772 15.0659 0.2057
0.8 105.357 0.5111 73.2423 0.4290 47.6494 0.3549 17.9011 0.2338
0.9 230.299 0.9633 141.697 0.7288 76.7823 0.5297 18.3979 0.2537
0.99 2467.38 9.1679 743.891 3.6115 89.5646 0.7829 73.7920 0.0692

5 0.7 451.313 9.0874 331.732 7.2409 232.916 5.6325 105.360 3.2388
0.8 652.082 12.7723 453.225 9.7249 294.798 7.1444 110.705 3.5839
0.9 1265.65 24.0712 778.849 16.5974 422.200 10.6691 101.254 3.8743
0.99 12401.6 229.079 3743.17 87.2753 451.618 17.9354 373.088 2.3922

10 0.7 1645.11 36.3492 1209.82 28.7364 850.020 22.1430 385.168 12.4717
0.8 2330.35 51.0881 1620.83 38.6357 1055.34 28.1486 397.402 13.8775
0.9 4431.27 96.2822 2729.89 66.0659 1482.56 42.2002 357.451 15.1433
0.99 42575.8 916.292 12885.5 348.721 1564.79 71.5612 1266.77 9.7050

200 1 0.7 51.8951 0.1592 42.1616 0.1490 33.5487 0.1391 20.2075 0.1207
0.8 85.2974 0.2235 66.6005 0.2040 50.4516 0.1855 26.9671 0.1516
0.9 185.026 0.4206 132.354 0.3597 89.3118 0.3039 35.7030 0.2093
0.99 1931.06 3.9945 801.767 2.0537 192.314 0.7878 13.5335 0.0631

5 0.7 265.637 3.9787 215.353 3.4238 170.926 2.9148 102.374 2.0546
0.8 384.132 5.5850 299.383 4.6468 226.286 3.8029 120.372 2.4415
0.9 745.649 10.5089 532.507 8.1126 358.563 6.0534 142.701 3.1030
0.99 7282.62 99.7940 3017.36 48.3717 720.906 16.6721 51.9463 1.2305

10 0.7 956.170 15.9145 774.997 13.5292 614.952 11.3592 368.095 7.7612
0.8 1354.06 22.3397 1055.12 18.3830 797.330 14.8522 423.951 9.2666
0.9 2571.73 42.0342 1836.38 32.1730 1236.35 23.7566 491.943 11.9066
0.99 24647.7 399.164 10211.6 193.054 2439.54 66.2922 174.788 4.9018

250 1 0.7 49.7937 0.1376 42.0037 0.1305 34.9404 0.1235 23.3224 0.1103
0.8 82.8689 0.1937 67.6386 0.1800 54.0985 0.1669 32.8819 0.1424
0.9 181.992 0.3651 138.031 0.3218 100.670 0.2813 48.7098 0.2099
0.99 1954.84 3.4733 927.931 1.9613 303.128 0.8999 18.7292 0.1124

5 0.7 266.658 3.4385 224.513 3.0327 186.357 2.6544 123.821 1.9913
0.8 389.032 4.8396 317.098 4.1467 253.220 3.5115 153.400 2.4393
0.9 761.738 9.1229 577.253 7.3240 420.579 5.7381 203.068 3.3072
0.99 7506.45 86.7781 3562.79 46.1198 1163.49 19.0945 71.3877 2.0149

10 0.7 948.340 13.7537 798.193 11.9834 662.298 10.3470 439.725 7.5317
0.8 1349.17 19.3579 1099.43 16.4008 877.710 13.7119 531.431 9.2588
0.9 2572.53 36.4902 1949.31 29.0336 1420.11 22.5058 685.605 12.6739
0.99 24732.5 347.098 11741.3 184.017 3836.20 75.8862 234.766 7.9755

change of OLS estimator to the proposed is about 73%. The
jackknife KL estimator performs better than the ridge and the
KL estimator[15]. However, when there is both multicollinear-
ity and outlier, the proposed RJKL study dominates other esti-

mators. The estimator with the closest performance is the Ro-
bust Kibria-Lukman estimator with about 8% difference. The
application result agrees with the theoretical proofs and the sim-
ulation. All the estimators produced the same regression sign.

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Table 8: Estimated MSEs of the estimators when there are 30% outliers and p=7

n σ̂ ρ α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
50 1 0.7 176.029 25.4476 117.630 20.5623 72.4841 16.2500 24.5439 9.6180

0.8 289.091 40.7835 180.230 31.6595 100.078 23.8026 26.7521 12.4852
0.9 632.508 87.9825 339.479 62.5223 146.664 41.7969 23.2801 16.3585
0.99 6842.66 927.927 1587.39 379.980 250.968 88.9143 789.588 5.9982

5 0.7 1198.50 249.099 799.409 198.493 491.401 154.303 165.777 88.1495
0.8 1738.52 356.258 1082.10 272.434 599.513 201.112 159.759 101.484
0.9 3389.34 676.860 1816.30 472.164 782.782 308.093 124.102 114.791
0.99 33298.5 6511.84 7712.37 2575.63 1228.32 565.161 3889.30 51.3169

10 0.7 4302.62 939.749 2870.54 747.659 1765.13 580.130 596.058 330.089
0.8 6101.25 1325.89 3799.03 1012.04 2106.00 745.378 562.091 374.207
0.9 11603.0 2464.02 6221.37 1714.74 2683.96 1115.34 426.552 412.784
0.99 111319.1 23366.9 25810.1 9187.00 4102.53 1989.79 12934.1 190.624

100 1 0.7 87.9633 0.7618 63.8612 0.6566 44.1094 0.5603 19.2165 0.3977
0.8 144.012 1.0742 98.6666 0.8876 62.8889 0.7205 22.5087 0.4541
0.9 314.994 2.0255 190.263 1.5195 100.081 1.0925 22.3053 0.5107
0.99 3387.24 19.2659 983.693 8.0896 111.755 2.0612 121.875 0.1558

5 0.7 628.585 19.0303 456.143 15.4409 314.869 12.2794 136.945 7.4362
0.8 907.147 26.8310 621.375 20.9103 395.976 15.8243 141.763 8.5224
0.9 1759.23 50.5780 1062.86 36.0667 559.477 24.2995 125.289 9.8913
0.99 17235.4 481.069 5012.56 198.963 576.233 49.5735 624.349 4.4673

10 0.7 2290.08 76.1193 1661.75 61.5544 1146.97 48.7586 498.577 29.2766
0.8 3245.87 107.318 2223.17 83.3977 1416.52 62.8993 506.765 33.6416
0.9 6171.17 202.300 3727.83 143.967 1961.68 96.7536 438.654 39.2026
0.99 59262.7 1924.14 17227.7 795.448 1975.93 198.076 2146.44 17.9635

200 1 0.7 73.1493 0.2996 58.8538 0.2771 46.2797 0.2555 27.1012 0.2157
0.8 120.479 0.4212 93.0287 0.3793 69.4843 0.3398 35.8860 0.2685
0.9 261.752 0.7935 184.570 0.6684 122.085 0.5546 46.3870 0.3661
0.99 2734.52 7.5420 1099.80 3.9736 242.901 1.6104 18.6298 0.1589

5 0.7 375.503 7.4839 301.842 6.4787 237.110 5.5537 138.592 3.9787
0.8 543.927 10.5212 419.764 8.8351 313.348 7.3109 161.745 4.8223
0.9 1057.87 19.8201 745.870 15.5525 493.402 11.8510 187.766 6.4108
0.99 10343.8 188.360 4164.78 96.1097 924.537 36.8939 71.6864 3.4590

10 0.7 1349.72 29.9343 1084.69 25.7389 851.829 21.8968 497.604 15.4259
0.8 1912.79 42.0830 1475.82 35.1269 1101.40 28.8672 568.271 18.7578
0.9 3636.66 79.2775 2563.59 61.9265 1695.46 46.9327 645.146 25.1003
0.99 34893.8 753.406 14047.6 383.982 3119.88 147.128 242.676 13.7813

250 1 0.7 77.3227 0.2738 65.2093 0.2573 54.2173 0.2414 36.1045 0.2114
0.8 128.843 0.3857 105.175 0.3550 84.1090 0.3256 51.0087 0.2711
0.9 283.208 0.7277 214.894 0.6342 156.734 0.5474 75.5029 0.3965
0.99 3040.22 6.9228 1442.93 4.0329 467.134 1.9628 26.1510 0.2998

5 0.7 408.258 6.8396 343.645 6.0789 285.092 5.3668 188.916 4.1066
0.8 597.276 9.6350 486.802 8.3489 388.581 7.1625 234.658 5.1307
0.9 1172.97 18.1740 888.968 14.8722 647.371 11.9286 310.650 7.2851
0.99 11587.6 172.879 5493.24 97.5796 1773.18 45.1740 98.5925 6.3387

10 0.7 1439.39 27.3570 1210.94 24.1521 1004.01 21.1663 664.426 15.9385
0.8 2051.01 38.5383 1670.75 33.1929 1332.80 28.2843 803.719 19.9655
0.9 3915.69 72.6924 2965.88 59.2099 2158.22 47.2351 1033.85 28.5155
0.99 37674.9 691.468 17842.1 389.808 5746.51 180.118 318.295 25.2059

Though, the intercept value of OLS estimator is the highest. We
noticed a sharp reduction in the intercept for other estimators,
especially the jackknife KL and the proposed.

6. Conclusion

We introduced a new robust estimator for the linear regression
model in this study and named it the Robust Jackknife Kibria

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Table 9: Regression coefficients and MSEs of estimator adopting the Hussein data.

Coefficients α̂ α̂M α̂K α̂M K α̂K L α̂M K L α̂JK L α̂RJK L
β0 208.8853 173.3445 178.6308 155.4823 148.3763 137.6202 92.9456 103.4135
β1 0.6130 0.9976 0.8627 1.1450 1.1124 1.2924 1.5698 1.5747
β2 1.2563 1.1153 1.1574 1.0569 1.0584 0.9985 0.8772 0.8867
β3 -1.2213 -1.1159 -1.2635 -1.1408 -1.3058 -1.1658 -1.3832 -1.2135
MSEs 1850.4816 864.4366 1353.9507 695.6977 935.0263 545.3211 785.1599 500.0968

Lukman (RJKL) Estimator. The RJKL estimator was proposed
to handle outlier and multicollinearity together. The estimator
was formed by grafting the M-estimator into the Jackknife Kib-
ria Lukman (JKL) estimator. We presented theorems that state
the necessary conditions for the new estimator to perform bet-
ter than the JKL estimator and other existing robust estimators
discussed. We observed a good performance in the simulation
study of the new estimator and the real-life data analysis. Both
results supports the efficiency of the new estimator.

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